352 8.4 Reaction, Diffusion, and Flow
The first term on the right-hand side of the equation is the diffusional term, with D being
the diffusion coefficient. The second term v can often be a complicated function that embodies
the reaction process. The form of the reaction–diffusion equation that describes Brownian
diffusion of a particle in the presence of an external force (the reaction element) is also known
as the Smoluchowski equation or Fokker–Planck equation (also sometimes referred to as the
Kolmogorov equation), all of which share essentially the same core features in the mathematical
modeling of Markov processes; as discussed previously in the context of localization microscopy
(see Chapter 4), these processes are memoryless or history independent. Here, the function v is
given by an additional drift drag component due to the action of an external force F on a particle
of viscous drag coefficient γ, and the Fokker–Planck equation in 1D becomes
(8.65)
∂
∂=
∂
∂
−
∂(
)
∂
P
t
D
P
x
FP
x
2
2
1
γ
An example of such a reaction–diffusion process in biology is the movement of molecular
motors on tracks, such as the muscle protein myosin moving along an F-actin filament, since
they can diffuse randomly along the 1D but experience an external force due to the power
stroke action of the myosin on the F-actin while a motor is in a given active state. However,
in addition to this mechanical component, there is also a chemical reaction component that
involves transitions into and out of a given active state of the motor. For example, if the
active state of a motor is labeled as i, and a general different state is j, then the full reaction–
diffusion equation becomes the Fokker–Planck equation plus the chemical transition kinetics
component:
(8.66)
∂
∂=
∂
∂
−
∂(
)
∂
+
−
(
)
∑
P
t
D
P
x
FP
x
k p
k p
i
i
j
ji
j
ij
i
2
2
1
γ
where kij and kji are the equivalent off-rates from state i to j, and on-rates from state j to i,
respectively. Most PDEs of reaction–diffusion equations are too complicated to be solved
analytically. However, the model of the process of a single molecular motor, such as myosin,
translocating on a track, such as F-actin, can be simplified using assumptions of constant
translocation speed v and steady-state probabilities. These assumptions are realistic in many
types of muscles, and they reduce the mathematical problem to a simple ordinary differential
equation (ODE) that can be solved easily. Importantly, the result compares very well with
experimentally measured dependence of muscle force with velocity of muscle contraction
(see Chapter 6).
The case of motion of molecular motor translocation on a track can be reduced to the
minimal model or reduced model, exemplified by myosin–actin. It assumes that there is just
one binding power stroke event between the myosin head and actin with rate constant kon
and one release step with rate constant koff. Assuming a distance d between accessible binding
sites on the actin track for a myosin head, a linear spring stiffness κ of the myosin–actin link
in the bound state (called the “crossbridge”) and binding of myosin to actin occur over a
narrow length range x0 to x0 + Δx, where the origin is defined as the location where the post–
power stroke crossbridge is relaxed, and release is assumed to only occur for x > 0. Under
these assumptions, the Fokker–Planck equation can be reduced to a simple ODE and solved
to give the average force per crossbridge of
(8.67)
〈〉=
−
−
F
d
k
x
v
x v
k
κ 1
2
0
2
2
2
exp
on
s
∆
An intuitive result from this model is that the higher the speed of translocation, the smaller
the average force exerted per crossbridge, which is an obvious conclusion on the grounds
of conservation of energy since there is a finite energy input due to the release of chemical
potential energy from the hydrolysis of ATP that fuels this myosin motor translocation (see
Chapter 2).